On continuous dependence in finite elasticity

QUARTERLY OF APPLIED MATHEMATICS 13$ APRIL 1980 ON CONTINUOUS DEPENDENCE IN FINITE ELASTICITY* By SCOTT J. SPECTOR (University of Tennessee) Abstract. We investigate the relation between stability and continuous dependence for a nonlinearly elastic body at equilibrium. We show that solutions of the governing equations that lie in a convex, stable set of deformations depend continuously on the body forces and the surface tractions. The definition of stability used is essentially due to Hadamard. 1. Introduction. In this note we consider the relation between stability and continuous dependence for a nonlinearly elastic body at equilibrium. We show that solutions of the governing equations that lie in a convex, uniformly Hadamard-stable set of deformations depend continuously on the body forces and surface tractions. Gurtin and Spector [1] have shown that Hadamard stability is also sufficient for uniqueness of solutions of the governing equations. Thus uniqueness and continuous dependence follow from the same assumptions. Hadamard stability of a set Q of deformations is the requirement that for some a > 0 f Vu ■/4(V/)V«> a||V«|L2 2(&) J n* for every deformation / in Q and every variation u. Here A is the elasticity tensor, the derivative of the stress response function with respect to the deformation gradient, while 96 is the region of space occupied by the body in a fixed reference configuration. 2. The response function. Stability. We consider a body 96 and we identify 96 with the properly regular1 region of R3 it occupies in a fixed reference configuration. Further, we denote by 3 and Sf complementary subsets of the boundary d96 with 3 non-empty and relatively open. A deformation / (of 96) will be a member of the space Def = {/e C\9I, IR3):det V/> 0}, while a variation u will be a smooth function u: 96 -* R3 which satisfies u = 0 on 3. Here det is the determinant and V the gradient operator in [R3.In particular, V/ is the tensor field with components (V/% = df/dxj. * Received February 26, 1979. The author especially wishes to thank M. E. Gurtin for his comments on a previous version of this manuscript. Thanks are also due to M. Aron and J. J. Roseman for correspondence concerning continuous dependence. 1 Cf. Fichera [2], p. 351. In particular, St is compact and has piecewise smooth (C1) boundary. 136 SCOTT J. SPECTOR We assume that the body is elastic with smooth response function S. Thus S(Vf(x),x) gives the (Piola-Kirchhoff) stress at any point xel when the body is deformed by f Writing F for V/(x) we define the elasticity tensor A(F, x): Lin -<■Lin by2 A(F,x) = dFS(F,x). (1) For convenience we write S(Vf) and A(Vf) for the fields on 0$ with values S(Vf(x), x) and A(Vf(x), x) respectively. Definition [1], A set Q <=Def is (uniformly) Hadamard-stable some a > 0 f or simply H-stable if for Vu ■A(Vf)Vu> a||Vu||2,w (2) for all f e Q and variations u, in which case we write stab(Q) for the largest a with this property. In the above inequality ||Vw||hm =| * Vm• Vw where for any T, V e Lin, V ' 7 = £ Vij'I'ij. i.j We will also have occasion to use the L2-norm of vector fields over 3d and y; \\U IL2(m■ = Jn U ■ U, \\u\\2L2(y) =| Jy U ■ U. Remark. The above notion of stability is essentially due to Hadamard3 [3, p. 252]. For a further discussion of stability cf. [1] and [4]. Gurtin and Spector [1] have shown that there is a neighborhood of the reference configuration which is uniformly H-stable if either: (a) the reference configuration is positive and natural4; or (b) S> —d3d and the reference configuration is homogeneous and strongly elliptic. 3. The mixed problem. Continuous dependence. The mixed problem (with dead loading) consists in finding a deformation/that satisfies: (i) the equation of virtual work f S(V/) -Wu=\s-u+\b-u J<a (3) for all variations u; and 2 We use Lin to denote the space of linear transformations from R3 into R3. 3 Hadamard stability is a static "criterion" for stability and its precise relation to dynamic stability is unclear. Eq. (2) cannot hold for all deformations, since global uniqueness would then follow. 4 A natural configuration whose elasticity tensor is positive definite when restricted to symmetric tensors. NOTES 137 (ii) the displacement boundary condition f = d on 3>. (4) Here, d e C°(S(, 1R3)is the surface deformation, s e IR3) the surface traction, and be R3) the body force. The triplet (d, s, b) will be referred to as the data, while a deformation /e Def that satisfies (3) and (4) will be called a solution corresponding to the data (d, s, b). Remark. The traction problem (Sf = d;M,Q) = 0) is excluded from our consideration since stab(H) is, in general, zero when 2 is empty. For a partial resolution of this problem cf. [5]. Remark. If s and b are continuous and / is a C2 solution corresponding to the data (d, s, b) then the divergence theorem can be used to show that (3) and (4) are equivalent to div S(V/) 4- b = 0 in @t, f = d on S(Wf)n= s on y. Note that the definition of stability is independent depend on 3>, the domain of d. Our main result is the following of the data although it does Theorem. Let Q be convex5 and H-stable. Then there exists a constant X > 0, which depends only on stab(fi) and gft, such that if / and / are solutions that lie in Q and correspond to data (d, s, b) and (d, s, 5) respectively, then \\f —f\\mm) ^ A(||s—s||L2(y>) + II6—&||l2(»))Remark. A direct consequence of our theorem is uniqueness: there is at most one solution corresponding to (d, s, b) in Q. On the other hand, there may be additional solutions which do not lie in Q and may therefore not depend continuously on the data. (For uniqueness under weaker hypotheses, cf. [1].) Proof of the theorem. Let Q c Def be convex and H-stable. Let u=f—f Then /=/ on S>, so that u = 0 on Q) and u is a variation; hence (3) yields f Vu•[S(V/) - S(V/)] = f u ■(s- s)+ f u ■(S- b). (5) Consider the function g: [0, 1] -►U defined by g{c)= f Vu • S(V/+ (7Vu). By (1) g'(a) = I Vu ■/4(V/+ a Vu) Vu. Hence, by the mean value theorem, gf(l) —g(0) = g'(£) at some e (0, 1); so that [ Vu • [S(V/)- S(V/)]= [ Vu • A(Wf+I Vu)Vu. * a* '' Convexity here is with respect to the linear structure in C'(, (6) 138 SCOTT J. SPECTOR Since Q is convex, (/ + <*u)e t) and therefore (2) gives us stab(n)||V«||21(a) < f Vu■A(Vf+HVw)V«. Next, by the Cauchy-Schwarz (7) inequality, (" u ■(b — b) < \\u\\L2m || fi — b||t2(s»), ^ , I U ■(s — s) < ||w||t2(^) ||s —s||/_2(y)• V (8) JSf Finally, the desired result follows from (5), (6), (7), (8) and the standard inequalities5 I|u||t2(<»)5S ||U||L2(y) ||V«||t2(a)) ^ ^l(||U||£J(») + ||Vu|| where > 0 depends only on 38. Remark. It is clear from the proof of the theorem that the L2-norm of (/ —/) can be replaced by the H'-norm: IMIhi<s9) = |ImIIl2(s») + ||^uIIl2(«)In addition, if we replace (8) by7 | u ■(b — b) < ||w||//i(<»)||—^||h-i(»)! I U • (s — s) < ||u||//i/2(y) ||s — ■s||//-i/2(y)» and use the trace theorem (cf. e.g. [6, p. 277]) IMIhi/2(y)^ ^2 ||M||//i(<ss)> we arrive at the stronger result ||/ —/||«i(ag) ^ ^(||s ~ s||//-i/2(y) + IIS —b\\H-1(m)). References [1] M. E. Gurtin and S. J. Spector, On stability and uniqueness in finite elasticity, Arch. Rat. Mech. Anal., to appear [2] G. Fichera, Existence theorems in elasticity, in Handbuch der Physik, VIa/2, Berlin, Springer-Verlag (1972) [3] J. Hadamard, Lemons sur la propagation des ondes et les equations d'hydrodynamique, Paris, Hermann (1903) [4] S. J. Spector, On uniqueness in finite elasticity with general loading, J. Elasticity, to appear [5] S. J. Spector, On uniqueness for the traction problem in finite elasticity, in preparation [6] R. A. Adams, Sobolev spaces, New York, Academic Press (1975) 6 The first is the Poincare inequality, cf., e.g., Fichera [2, p. 274, footnote 17], whose proof with minor modification applies in the present circumstance. The second is the trace theorem, cf., e.g., Fichera [2, p. 353], Adams [6, p. 113]. 7 The negative and fractional norms are defined in the usual manner. Cf., e.g., [6].